The rotation of molecules

It must be pointed out that another type of internal motion is the overall rotation of the molecule. The vibration and rotation of the molecule are shown schematically in figure Al.2.2. 

Figure Al.2.2. Internal nuclear motions of a diatomic molecule. Top the molecule in its equilibrium configuration. Middle vibration of the molecule. Bottom rotation of the molecule.

At this point the reader may feel that we have done little in the way of explaining molecular synnnetry. All we have done is to state basic results, nonnally treated in introductory courses on quantum mechanics, connected with the fact that it is possible to find a complete set of simultaneous eigenfiinctions for two or more commuting operators. However, as we shall see in section Al.4.3.2. the fact that the molecular Hamiltonian //coimmites with and F is intimately coimected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing tlirough the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the... 

We consider rotations of the molecule about space-fixed axes in the active picture. Such a rotation causes the (x, y, z) axis system to rotate so that the Euler angles change... 

When the molecule is not in a S state there is an interaction between the rotation of the molecule and S and/or L, and the details of coupling the angular momenta are involved. Most nonsinglet molecules with electronic orbital angular momentum A = 0 obey Hund s case (b) coupling. In Case (b), the electronic orbital angular momentum combines with the nuclear orbital angular... 

Determination of the Permutation Descriptor after Rotation of the Molecule... 

Corresponding to every symmetry element is a symmetry operation which is given the same symbol as the element. For example, C also indicates the actual operation of rotation of the molecule by 2n/n radians about the axis. 

Molecules initially in the J = 0 state encounter intense, monochromatic radiation of wavenumber v. Provided the energy hcv does not correspond to the difference in energy between J = 0 and any other state (electronic, vibrational or rotational) of the molecule it is not absorbed but produces an induced dipole in the molecule, as expressed by Equation (5.43). The molecule is said to be in a virtual state which, in the case shown in Figure 5.16, is Vq. When scattering occurs the molecule may return, according to the selection mles, to J = 0 (Rayleigh) or J = 2 (Stokes). Similarly a molecule initially in the J = 2 state goes to... 

C , rotation of the molecule about a symmetry axis through an angle of 360°/n n is called the order of the rotation (twofold, threefold, etc.) ... 

S , Rotation of the molecule through an angle 360°/n followed by reflection of all atoms through a plane perpendicular to the axis of rotation the combined operation (which may equally follow the sequence reflection then rotation) is called improper rotation-,... 

Once the electronic Schrodinger equation has been solved for a large number of nuclear geometries (and possibly also for several electronic states), the PES is known. This can then be used for solving the nuclear part of the Schrodinger equation. If there are N nuclei, there are 3N coordinates that define the geometry. Of these coordinates, three describe the overall translation of the molecule, and three describe the overall rotation of the molecule with respect to three axes. Eor a linear molecule, only two coordinates are necessary for describing the rotation. This leaves 3N-6(5) coordinates to describe the internal movement of the nuclei, the vibrations, often chosen to be... 

This energy increase can take different forms. It can be added as translational kinetic energy to speed up the movement to and fro of the molecules it can be added to the rotations of the molecules to get them to spin faster it can be added to increase the amplitude of the vibrational oscillations of the molecules and it can be added to excite electrons to higher energy states in the atoms or molecules. Other forms of internal energy are also possible, but the above are the most common. 

As the density of a gas increases, free rotation of the molecules is gradually transformed into rotational diffusion of the molecular orientation. After unfreezing , rotational motion in molecular crystals also transforms into rotational diffusion. Although a phenomenological description of rotational diffusion with the Debye theory [1] is universal, the gas-like and solid-like mechanisms are different in essence. In a dense gas the change of molecular orientation results from a sequence of short free rotations interrupted by collisions [2], In contrast, reorientation in solids results from jumps between various directions defined by a crystal structure, and in these orientational sites libration occurs during intervals between jumps. We consider these mechanisms to be competing models of molecular rotation in liquids. The only way to discriminate between them is to compare the theory with experiment, which is mainly spectroscopic. 

Other arrangements may he drawn, but they differ from the one shown only by a rotation of the molecule. 

The rotation of the ammonium ion in salts at ordinary temperatures provides justification for the customary treatment of the ion as spherically symmetrical in the theoretical discussion of the structure of ionic crystals. Further, the rotation of molecules such as NHj and H20 about symmetry axes accounts for the fact that these molecules occupy positions in crystals with symmetry elements not compatible with those of the non-rotating molecule. Thus in Ni(NH3)6CU the NHj molecules lie on four-fold axes, and in alum the H2O molecules on three-fold axes. The rotation of the molecules,... 

Fig. 139.—The free-draining molecule during viscous flow of the solution. Flow of the solvent relative to the center of gravity (X) of the molecule and the rotation of the molecule as a whole are indicated.

Figure 1.10 UHV-STM images of HtB-HBC adsorption on Cu l 1 0 showing the correlation between the orientation of the adsorbate lattice vectors and the local rotation of the molecule away from its high symmetry azimuthal orientation atop a copper atom.

The results are insensitive to rotation of the molecule in its plane, as long as the molecular centre is kept fixed above the central Ni atom, at a distance of 2 A. The shift of the ethylene Tf-level is correctly calculated. 

Atomic units (me = 1, qe = 1, h = 1) are used throughout this chapter.] The coefficients T, T2, and To are assumed to be in general analytical functions of the bending coordinate p. The term Tz represent the operator describing the rotation of the molecule around the (principal) axis z corresponding to the smallest moment inertia—this axis coincides at the linear nuclear arrangement with the molecular axis. Now Tz can be written in the form... 

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